To obtain the \(L^2_{t,x}\) -estimate of \(a_{(k)}\) , \(k\in \Lambda _{B}\) ,
we use () and () to bound
\(\Vert a_{(k)}\Vert _{L^2_{t,x}}&\le \Vert  \rho _B \Vert _{L^1_{t,x}}^{\frac{1}{2}} \Vert  \gamma _{(k)} \Vert _{C_x(B_{\varepsilon _B}(0))} \Vert f_B\Vert _{C_t} \\&\le M_* (8 \varepsilon _B^{-1})^{\frac{1}{2}}(16\pi ^4 \delta _{q+1} + \Vert \mathring{R}_{\ell }^{B} \Vert _{L^1_{t,x}})^{\frac{1}{2}} \\&\le M_* (8 \varepsilon _B^{-1})^{\frac{1}{2}}( 16\pi ^4 + 1)^{\frac{1}{2}} \delta _{q+1}^{\frac{1}{2}},\) 

which yields (REF ).
Moreover, for \(N\ge 1\) ,
by () and the standard Hölder estimate (cf. [1]}),
\(\Vert a_{(k)}\Vert _{C^N_{t,x}}&\lesssim \sum _{0\le N_1+N_2+N_3\le N}\Vert \rho _B^{\frac{1}{2}} \Vert _{C^{N_1}_{t,x}}\Vert  \gamma _{(k)} (-\rho _B^{-1} \mathring{R}^B_{\ell } ) \Vert _{C^{N_2}_{t,x}} \Vert f_B\Vert _{C_t^{N_3}} \\&\lesssim \sum _{0\le N_1+N_2+N_3\le N} \ell ^{-N_1-1}\ell ^{-N_3}\left(1+ \Vert \rho _B^{-1}\mathring{R}^B_{\ell }\Vert _{C^{N_2}_{t,x}} + \Vert \rho _B^{-1}\mathring{R}^B_{\ell }\Vert _{C_{t,x}}^{N_2-1}\Vert \rho _B^{-1}\mathring{R}^B_{\ell }\Vert _{C^{N_2}_{t,x}}\right) .\) 

Since by (), () and (),
\(\Vert \rho _B^{-1}\mathring{R}^B_{\ell }\Vert _{C^{N_2}_{t,x}}&\lesssim \sum _{0\le N_{21}+N_{22} \le N_2}\Vert \rho _B^{-1}\Vert _{C^{N_{21}}_{t,x}}\Vert \mathring{R}^B_{\ell }\Vert _{C^{N_{22}}_{t,x}} \nonumber \\& \lesssim \sum _{0\le N_{21}+ N_{22}\le N_2}\ell ^{-N_{21}-1}\ell ^{-N_{22}}\lesssim \ell ^{-N_{2}-1},\) 

we get
\(\Vert a_{(k)}\Vert _{C^N_{t,x}} &\lesssim \sum _{0\le N_1+N_2+N_3\le N}\ell ^{-N_1-1}\ell ^{-N_3}(\ell ^{-N_2-1}+\ell ^{-2N_2} )\lesssim \ell ^{-4N}.\) 
The \(C_{t,x}\) -estimate of \(a_{(k)}\)  can be bounded by,
via (),
\(\Vert a_{(k)}\Vert _{C_{t,x}}&\lesssim \Vert  \rho _B^{\frac{1}{2}} \Vert _{C_{t,x}}\Vert f_B\Vert _{C_t} \Vert  \gamma _{(k)} \Vert _{C_x(B_{\varepsilon _B}(0))}\lesssim \ell ^{-1}.\) 

Therefore, the estimates in () are verified.
The proof is complete.
